Oscar Zariski was one of the most influential mathematicians working in the filed of algebraic geometry in the twentieth century. He was born as Ascher Zaritsky on 24 April 1899, in Kobrin (now in Belarus, then in Russia) in a Jewish family.
He was a student at the University of Kiev in 1918, moving to Rome to study in 1920. He became a disciple of the Italian school of algebraic geometry, studying with Guido Castelnuovo, Federigo Enriques and Francesco Severi. He wrote a doctoral dissertation in 1924, on a topic in Galois theory. It was when it came to be published that he accepted a suggestion to change his name for professional purposes.
It was this period that he wrote the celebrated book Algebraic Surfaces, intended as a summation of the work of the Italian school, but in effect its swansong, too. It was published in 1935. It was reissued many years later, with copious notes showing how much the field of algebraic geometry had changed, not only foundationally but in emphasis. It is still an important reference.
It seems to have been this work that set the seal of Zariski's discontent with the approach of the Italians to birational geometry. The question of rigour he addressed by recourse to commutative algebra. The Zariski topology, as it was later known, is adequate for biregular geometry, where varieties are mapped by polynomial functions. That theory is too limited for algebraic surfaces, and even for curves with singular points. A rational map is to a regular map as a rational function is to a polynomial: it may be indeterminate at some points. In geometric terms, one has to work with functions defined on some open, dense set of a variety V. The description of the behaviour on the complement may require infinitely near points to be introduced to account for limiting behaviour along different directions. This introduces a need, in the surface case, to use also valuation theory to describe the phenomena such as 'blowing up' (balloon-style, rather than explosively).
Zariski became professor at Harvard University in 1947, retiring in 1969. In 1945 he fruitfully discussed foundational matters for algebraic geometry with André Weil; in fact Weil's interest was in putting an abstract variety theory in place, to support the use of the Jacobian variety in his proof of the Riemann hypothesis for curves over finite fields, a direction rather oblique to Zariski's interests. The two sets of foundations weren't reconciled, at that point.
At Harvard, Zariski's students included Shreeram Abhyankar, Heisuke Hironaka, David Mumford and Michael Artin - thus spanning the main areas of advance in singularity theory, moduli theory and cohomology in the next generation. Zariski himself worked on equisingularity theory. Some of his major results, Zariski's main theorem and the Zariski theorem on holomorphic functions, were amongst the results generalized and included in the programme of Alexander Grothendieck that ultimately unified algebraic geometry.
He was awarded the Steele Prize in 1981. He wrote also Commutative Algebra in two volumes, with Pierre Samuel. His papers have been published by MIT Press, in four volumes. The Unreal Life of Oscar Zariski (1991) is a biography by Carol Ann Parikh.